PATH

This is part of the mapping module | |

It is only available if you configure PLUMED with ./configure –enable-modules=mapping . Furthermore, this feature is still being developed so take care when using it and report any problems on the mailing list. |

Path collective variables with a more flexible framework for the distance metric being used.

The Path Collective Variables developed by Branduardi and co-workers [24] allow one to compute the progress along a high-dimensional path and the distance from the high-dimensional path. The progress along the path (s) is computed using:

\[ s = \frac{ \sum_{i=1}^N i \exp( -\lambda R[X - X_i] ) }{ \sum_{i=1}^N \exp( -\lambda R[X - X_i] ) } \]

while the distance from the path (z) is measured using:

\[ z = -\frac{1}{\lambda} \ln\left[ \sum_{i=1}^N \exp( -\lambda R[X - X_i] ) \right] \]

In these expressions \(N\) high-dimensional frames ( \(X_i\)) are used to describe the path in the high-dimensional space. The two expressions above are then functions of the distances from each of the high-dimensional frames \(R[X - X_i]\). Within PLUMED there are multiple ways to define the distance from a high-dimensional configuration. You could calculate the RMSD distance or you could calculate the amount by which a set of collective variables change. As such this implementation of the path CV allows one to use all the difference distance metrics that are discussed in Distances from reference configurations. This is as opposed to the alternative implementation of path (PATHMSD) which is a bit faster but which only allows one to use the RMSD distance.

The \(s\) and \(z\) variables are calculated using the above formulas by default. However, there is an alternative method of calculating these collective variables, which is detailed in [38]. This alternative method uses the tools of geometry (as opposed to algebra, which is used in the equations above). In this alternative formula the progress along the path \(s\) is calculated using:

\[ s = i_2 + \textrm{sign}(i_2-i_1) \frac{ \sqrt{( \mathbf{v}_1\cdot\mathbf{v}_2 )^2 - |\mathbf{v}_3|^2(|\mathbf{v}_1|^2 - |\mathbf{v}_2|^2) } }{2|\mathbf{v}_3|^2} - \frac{\mathbf{v}_1\cdot\mathbf{v}_3 - |\mathbf{v}_3|^2}{2|\mathbf{v}_3|^2} \]

where \(\mathbf{v}_1\) and \(\mathbf{v}_3\) are the vectors connecting the current position to the closest and second closest node of the path, respectfully and \(i_1\) and \(i_2\) are the projections of the closest and second closest frames of the path. \(\mathbf{v}_2\), meanwhile, is the vector connecting the closest frame to the second closest frame. The distance from the path, \(z\) is calculated using:

\[ z = \sqrt{ \left[ |\mathbf{v}_1|^2 - |\mathbf{v}_2| \left( \frac{ \sqrt{( \mathbf{v}_1\cdot\mathbf{v}_2 )^2 - |\mathbf{v}_3|^2(|\mathbf{v}_1|^2 - |\mathbf{v}_2|^2) } }{2|\mathbf{v}_3|^2} - \frac{\mathbf{v}_1\cdot\mathbf{v}_3 - |\mathbf{v}_3|^2}{2|\mathbf{v}_3|^2} \right) \right]^2 } \]

The symbols here are as they were for \(s\). If you would like to use these equations to calculate \(s\) and \(z\) then you should use the GPATH flag. The values of \(s\) and \(z\) can then be referenced using the gspath and gzpath labels.

- Examples

In the example below the path is defined using RMSD distance from frames.

Click on the labels of the actions for more information on what each action computes